Let {ak} be a sequence. Prove Theorem 7.6 (a) along with the...

Let $\left\{a_{k}\right\}$ be a sequence. Prove Theorem $7.6$ (a) along with the following variations: (a) Show that when $a_{k+1}-a_{k} \geq 0$ for every $k \geq 1$, the sequence is increasing. (b) Show that when $a_{k+1}-a_{k}>0$ for every $k \geq 1$, the sequence is strictly increasing. (c) Show that when $a_{k+1}-a_{k} \leq 0$ for every $k \geq 1$, the sequence is decreasing. (d) Show that when $a_{k+1}-a_{k}<0$ for every $k \geq 1$, the sequence is strictly decreasing.

Let $\left\{a_{k}\right\}$ be a sequence. Prove Theorem $7.6$ (a) along with the following variations:
(a) Show that when $a_{k+1}-a_{k} \geq 0$ for every $k \geq 1$, the sequence is increasing.
(b) Show that when $a_{k+1}-a_{k}>0$ for every $k \geq 1$, the sequence is strictly increasing.
(c) Show that when $a_{k+1}-a_{k} \leq 0$ for every $k \geq 1$, the sequence is decreasing.
(d) Show that when $a_{k+1}-a_{k}<0$ for every $k \geq 1$, the sequence is strictly decreasing.

Key Concepts

Order Properties of Real Numbers

The order properties of real numbers ensure that inequalities behave predictably under addition and subtraction. These properties are essential when working with inequalities like a??? - a? ? 0 or a??? - a? > 0, as they allow one to infer the overall ordering of a sequence from the behavior of individual differences between consecutive terms.

Monotonic Sequences

A monotonic sequence is one that consistently maintains either a non-decreasing or non-increasing order throughout its terms. In the context of this question, understanding monotonic sequences allows one to conclude that if the difference between consecutive terms is non-negative (or non-positive), then the entire sequence follows an increasing (or decreasing) pattern.

Strict Monotonicity

Strict monotonicity refers to sequences where each term is strictly larger or strictly smaller than its predecessor. This stronger condition, demonstrated when the difference between consecutive terms is strictly positive or strictly negative, eliminates the possibility of equal successive terms and guarantees a unique progression.

Consecutive Differences

The consecutive differences method involves analyzing the expression a??? - a? to determine a sequence's behavior. By examining whether this difference is non-negative, strictly positive, non-positive, or strictly negative, one can directly conclude whether the sequence is increasing, strictly increasing, decreasing, or strictly decreasing.

Proof by Mathematical Induction

Mathematical induction is a powerful proof technique used to confirm that a given property holds for all natural numbers. In proving the monotonicity of sequences, induction is typically used to show that if the property holds for one term (or a base case), then it necessarily holds for the subsequent term, thereby extending the result to all terms in the sequence.

Transcript

Please give Ace some feedback